## The Annals of Statistics

- Ann. Statist.
- Volume 46, Number 1 (2018), 247-279.

### High-dimensional asymptotics of prediction: Ridge regression and classification

Edgar Dobriban and Stefan Wager

#### Abstract

We provide a unified analysis of the predictive risk of ridge regression and regularized discriminant analysis in a dense random effects model. We work in a high-dimensional asymptotic regime where $p,n\to\infty$ and $p/n\to\gamma>0$, and allow for arbitrary covariance among the features. For both methods, we provide an explicit and efficiently computable expression for the limiting predictive risk, which depends only on the spectrum of the feature-covariance matrix, the signal strength and the aspect ratio $\gamma$. Especially in the case of regularized discriminant analysis, we find that predictive accuracy has a nuanced dependence on the eigenvalue distribution of the covariance matrix, suggesting that analyses based on the operator norm of the covariance matrix may not be sharp. Our results also uncover an exact *inverse* relation between the limiting predictive risk and the limiting estimation risk in high-dimensional linear models. The analysis builds on recent advances in random matrix theory.

#### Article information

**Source**

Ann. Statist., Volume 46, Number 1 (2018), 247-279.

**Dates**

Received: December 2015

Revised: November 2016

First available in Project Euclid: 22 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1519268430

**Digital Object Identifier**

doi:10.1214/17-AOS1549

**Mathematical Reviews number (MathSciNet)**

MR3766952

**Zentralblatt MATH identifier**

06865111

**Subjects**

Primary: 62H99: None of the above, but in this section

Secondary: 62J05: Linear regression 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]

**Keywords**

High-dimensional asymptotics ridge regression regularized discriminant analysis prediction error random matrix theory

#### Citation

Dobriban, Edgar; Wager, Stefan. High-dimensional asymptotics of prediction: Ridge regression and classification. Ann. Statist. 46 (2018), no. 1, 247--279. doi:10.1214/17-AOS1549. https://projecteuclid.org/euclid.aos/1519268430

#### Supplemental materials

- Supplement to “High-dimensional asymptotics of prediction: Ridge regression and classification”. In the supplementary material, we give efficient methods to compute the risk formulas, and prove the remaining lemmas and other results.Digital Object Identifier: doi:10.1214/17-AOS1549SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.